Process matrices are classified in terms of the degree of causal structure that they exhibit, as shown in figure REF . First, we have the process matrices which are causally ordered, i.e. they are compatible with the existence of a strict partial order such that the probabilities for the outcomes in some laboratory depend only on the settings in that laboratory or in experiments which precede it in the partial order. Note that here and throughout this article, `depend' refers to statistical dependence, i.e. it encodes the possibility of signalling - the kinds of relationships that may occur between no-signalling laboratories, as in the Bell correlations, do not count for the purpose of these definitions. Second, we have process matrices which are causally separable, i.e. they can be written as a convex decomposition of causally ordered process matrices. Third, we have process matrices which are neither causally ordered nor causally separable, but are still causal in the sense that we can write down a well-defined probability distribution over possible strict partial orders such that in each possible case, the choice of setting in a local experiment does not affect the occurrence of events which are earlier in the order, nor the strict partial order on the set of the events and the experiment in question[1]}.More precisely, according to ref [1]} a process matrix is causal iff there exists a probability distribution over possible strict partial orders on the set of local experiments and outcomes of those experiments such that for every local experiment \(A\) , every subset \(X\)  of the other local experiments, and every choice of strict partial order, the joint probability distribution over the outcomes for the experiments \(X\)  and the possible strict partial orders over \(A, X\)  (obtained by summing over only those orders for the full process such that \(A\)  does not precede any experiments in \(X\) ) are independent of the settings in the experiment \(A\) . Finally, we have the process matrices which are non-causal, meaning that they are not compatible with a distribution over strict partial orders which obeys these requirements. We can define `causal inequalities,' somewhat analogous to the Bell inequalities which can only be violated by non-causal processes[3]}. Although a number of non-causal process matrices have been formulated and studied, all processes currently known to be realised in nature (without post-selection) are causal.
